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HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2008 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Section 6.1
Introduction to the Normal
Curve
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Normal Distribution:
A continuous probability distribution for a given
random variable, X, that is completely defined by it’s
mean and standard deviation.
Properties of a Normal Distribution:
1. A normal curve is symmetric and bell-shaped.
2. A normal curve is completely defined by its mean,
, and standard deviation, .
3. The total area under a normal curve equals 1.
4. The x-axis is a horizontal asymptote for a normal
curve.
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Symmetric and Bell-Shaped:
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Completely Defined by its Mean and Standard Deviation:
An inflection point is a point on the curve where the curvature of the
line changes. The inflection points are located at  - and  + .
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Total Area Under the Curve = 1:
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
The x-Axis is a Horizontal Asymptote:
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Determine if the following is a normal distribution:
a. Birth weights of 75 babies.
Normal
b. Ages of 250 students in 10th grade.
No, this would be uniform
c. Heights of 100 adult males.
Normal
d. Frequency of outcomes from rolling a
die.
No, because the data is discrete
e. Weights of 50 fully grown tigers.
Normal
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
How Many Normal Curves are there?
Because there are an infinite number of possibilities for  and ,
there are an infinite number of normal curves.
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Standard Normal Distribution:
A standard normal distribution has the same
properties as the normal distribution; in addition, it has
a mean of 0 and a standard deviation of 1.
Properties of a Standard Normal Distribution:
1. The standard normal curve is symmetric and bellshaped.
2. It is completely defined by its mean and standard
deviation,  = 0 and  = 1.
3. The total area under a standard normal curve
equals 1.
4. The x-axis is a horizontal asymptote for a standard
normal curve.
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Converting to the Standard Normal Curve:
Standard Score Formula (z-score):
When calculating the z-score, round your answers to two
decimal places.
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Draw a Normal Curve:
Given  = 40 and  = 5, indicate the mean, each of the
inflections points, and where each given value of x will
appear on the curve.
x1 = 33 and x2 = 51
Solution:
35
33
40
45
51
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Convert to the Standard Normal Curve:
Given  = 40 and  = 5, calculate the standard score
for each x value and indicate where each would appear
on the standard normal curve.
x1 = 33 and x2 = 51
Solution:
-1
-1.4
0
1
2.2
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.1 Introduction to the Normal
Curve
Convert to the Standard Normal Curve:
Given  = 48 and  = 5, convert to a normal curve and
indicate where a score of x = 45 would appear on each
standard normal curve.
Solution:
43
45
48
53
-1
-0.6
0
1